I will describe certain geometric, topological, and analytical aspects of M-theory on manifolds with boundaries and with corners of codimension two. The study of the fields and their partition functions involves, via index theory, certain primary, secondary and tertiary invariants. From a physical point of view, one has a hierarchy of theories of the gauge, Chern-Simons, and WZW types, respectively. From the chromatic point of view, these correspond to cohomology, K-theory, and elliptic cohomology. Some aspects of the above can be globalized using certain moduli n-stacks.